Bregman Proximal Langevin Monte Carlo via Bregman-Moreau Envelopes

Tim Tsz Kit Lau, Han Liu*

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

Abstract

We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such algorithms leveraging recent advances in convex analysis and optimization methods involving Bregman divergences, namely the Bregman-Moreau envelopes and the Bregman proximity operators, and in the Langevin Monte Carlo algorithms reminiscent of mirror descent. The proposed algorithms extend existing Langevin Monte Carlo algorithms in two aspects-the ability to sample nonsmooth distributions with mirror descent-like algorithms, and the use of the more general Bregman-Moreau envelope in place of the Moreau envelope as a smooth approximation of the nonsmooth part of the potential. A particular case of the proposed scheme is reminiscent of the Bregman proximal gradient algorithm. The efficiency of the proposed methodology is illustrated with various sampling tasks at which existing Langevin Monte Carlo methods are known to perform poorly.

Original languageEnglish (US)
Pages (from-to)12049-12077
Number of pages29
JournalProceedings of Machine Learning Research
Volume162
StatePublished - 2022
Event39th International Conference on Machine Learning, ICML 2022 - Baltimore, United States
Duration: Jul 17 2022Jul 23 2022

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

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